I. A. Ender

Standard Moment Method in the Problems on Ion Kinetics in Neutral Gas

A new method for calculating matrix elements of the collision integral is presented. This method is applied both for the determination of ion mobility on atom background and the construction of the ion distribution function in the case when an electric field or crossed electric and magnetic fields are switched on instantaneously. The limits of the nonstationary moment method are explored. Five interaction models are studied in detail, and the evolution of the distribution function is presented for various electric field strengths. It is shown that the angle and energy dependence of the crosssection strongly influences the distribution especially in the high velocity domain. In particular, the runaway of ions for the Coulomb interaction is determined. Moreover, the mobility vector is shown to move along a spiral to its stationary value when an electric and cross magnetic field are switched on.

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels Ll(v, v1) of linear collision integral and kernels Gl, 0l(v, v1, v2) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels Gl, 0+l(v, v1, v2) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.

Relaxation of a hard-sphere gas and criteria of validity of the calculations

The isotropic Boltzmann equation is solved using the Sonin moment system of equations. The main task here is the construction of a matrix describing the particle interaction. Such a matrix has been constructed analytically for various interaction cross sections in a number of papers [M. Barnsley and G. Turchetti, Lett. Nuovo Cimento 33, 347 (1982); G. Turchetti and M. Paolilli, Phys. Lett. A 90, 123 (1982); F. Schürrer and G. Kügerl, Phys. Fluids A 2, 609 (1990); A. Ya. Énder and I. A. Énder, Tech. Phys. 39, 997 (1994)]. Calculations of matrix elements arrived at by different methods for the hard-sphere model are compared. Some general properties of the matrix are found that can be used as criteria of the validity of the calculations. With the help of such criteria it is shown that the nonlinear matrix elements were calculated incorrectly by Schürrer and Kügerl (op. cit.).

Numerical solution of isotropic relaxation problems by the method of Maxwellian expansion

Abstract A numerical analytic method of solving the Boltzmann equation based on Maxwell expanding the distribution function and the collision integral is perfected. Two test problems are solved by this method: the temperature relaxation of pseudo-Maxwellian molecules on a Maxwellian background, and the non-linear relaxation of the sum of two Maxwellian distributions. The effectiveness of the numerical scheme is demonstrated. Different varieties of non-monotonic relaxation are investigated.

Polynomial expansion for the axially symmetric Boltzmann equation and relation between matrix elements of collision integral

Polynomial expansions are of widespread use in a gas kinetic theory. For linearized Boltzmann equation, such an expansion is a basis of the wide-known Enskog-Chapman method [1], [2]. For the nonlinear case, this method was developed in Burnetts [3] and Grads [4] works. As it was proved by Kumar [5], the best time-consuming expansion is that over so called spherical Hermite polynomials. Along Burnetts treatment, we use the real non-normalised spherical Hermite polynomials

Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory.

Calculation of boltzmann equation collision integral and accurate solution of relaxation problem

A method of solving the Boltzmann equation on the basis of expansion in Maxwell distributions (Maxwellians) is developed. The calculation of the collision integral consists of two stages: calculation of a two-Maxwellian collision integral and its subsequent expansion in Maxwellians. A special numerical procedure is proposed for approximating functions of one and two variables by a finite number of exponentials; the mean-square deviation of the accurate and approximate functions is minimized and certain moments of these functions are brought into agreement. For a solid-sphere model, the two-Maxwellian collision integral comprises a set of 24 δ functions over the whole range of the parameters investigated. The Euler method is used to examine the equalization of velocities and temperatures in the gas (“quasishock”) and resonance recharging of ions on atoms. The accuracy of the solutions obtained is estimated.

Boltzmann equation in the α, u representation

It is proposed, as in [1, 2], to regard the distribution function as a set of all possible Maxwellian distributions with arbitrary temperatures and mean velocities, with each Maxwelian distribution taken with its own weight ϕ (α, u). An equation equivalent to the Boltzmann equation is constructed for this weight function. The cases of one-dimensional, twodimensional, operator in the arbitrary case in the α, u representation it suffices to know the corresponding kernel for the one-dimensional problem.